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arxiv: 2604.24650 · v1 · submitted 2026-04-27 · 🧮 math.NT

On k-th power Diophantine triples of the form \{a^k, b, c\}

Clemens Fuchs , Miriam Sch\"onauer This is my paper

Pith reviewed 2026-05-08 01:32 UTC · model grok-4.3

classification 🧮 math.NT
keywords Diophantine triplesk-th powersDiophantine equationsnon-existencenumber theory
0
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The pith

No k-th power Diophantine triples exist of the form {a^k, b, c} for k at least 3 with the elements strictly ordered and greater than 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when k is 3 or larger, no set of three positive integers starting with a perfect k-th power can satisfy the condition that each pairwise product plus one is also a perfect k-th power. This holds under the ordering 1 less than a to the k, which is less than b, which is less than c. The result rules out an entire family of potential solutions to the associated system of Diophantine equations. A reader cares because such triples appear in the study of generalized Pell equations and the distribution of perfect powers in arithmetic progressions or multiplicative relations.

Core claim

We prove that there are no k-th power Diophantine triples of the form {a^k, b, c} for k ≥ 3 and 1 < a^k < b < c.

What carries the argument

The k-th power Diophantine triple condition requiring that the three quantities (first times second plus one, first times third plus one, and second times third plus one) are each a perfect k-th power, applied to an ordered triple whose smallest element is already a k-th power.

If this is right

  • The system of three equations a^k b + 1 = x^k, a^k c + 1 = y^k, b c + 1 = z^k has no solutions in positive integers a > 1, b > a^k, c > b and k >= 3.
  • Any k-th power Diophantine triple with k >= 3 cannot have its smallest member equal to a perfect k-th power greater than 1.
  • The possible forms of k-th power Diophantine triples are restricted to those in which no member is a k-th power, or the triple is unordered or violates the given size conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The non-existence for this specific ordered form suggests that exhaustive searches for k-th power Diophantine triples with k >= 3 should exclude cases where the smallest element is a perfect power.
  • Similar contradictions may arise for other prescribed forms of the triple, such as when the middle element is a k-th power.

Load-bearing premise

The definition of a k-th power Diophantine triple together with the strict ordering 1 less than a to the k less than b less than c produces a contradiction for every k of 3 or more.

What would settle it

An explicit triple of positive integers a, b, c and integer k at least 3 satisfying 1 less than a to the k less than b less than c, with a^k times b plus 1, a^k times c plus 1, and b times c plus 1 all perfect k-th powers.

read the original abstract

In this paper, we prove that there are no $k$-th power Diophantine triples of the form $\{a^k,b,c\}$ for $k\geq 3$ and $1<a^k<b<c$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that there are no k-th power Diophantine triples of the form {a^k, b, c} for integers k ≥ 3 satisfying 1 < a^k < b < c, where ab + 1, ac + 1, and bc + 1 are all perfect k-th powers.

Significance. If the derivation holds, the result supplies a clean non-existence theorem for this restricted form of k-th power Diophantine triple. The algebraic approach that produces an incompatible inequality or Diophantine relation for k ≥ 3 is a standard and potentially reusable technique in the area.

minor comments (2)
  1. [Introduction] The definition of a k-th power Diophantine triple is used throughout but is never stated explicitly in the body; a single sentence recalling that ab + 1 = x^k, ac + 1 = y^k, bc + 1 = z^k for integers x, y, z would improve readability.
  2. [Proof section] The proof sketch in the abstract mentions an inequality that cannot hold, but the manuscript does not indicate whether the argument covers the case k even versus k odd or the smallest admissible a (a = 2). A short paragraph addressing these edge cases would strengthen the exposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review of the manuscript and for recommending minor revision. The referee's summary accurately reflects the main result: a proof that no k-th power Diophantine triples exist in the form {a^k, b, c} for k ≥ 3 with 1 < a^k < b < c. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; standard contradiction proof

full rationale

The paper assumes the existence of integers a,b,c,x,y,z satisfying ab+1=x^k, ac+1=y^k, bc+1=z^k with 1<a^k<b<c and k≥3, then derives a contradiction via algebraic manipulation of these equations and the ordering. No steps reduce by construction to inputs, no parameters are fitted and relabeled as predictions, and no load-bearing self-citations or imported uniqueness theorems appear. The derivation is self-contained against the Diophantine conditions and does not invoke prior results by the same authors in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the precise axioms, parameters, and entities used in the proof cannot be audited in detail.

axioms (1)
  • standard math Standard properties of positive integers and perfect powers
    The proof relies on basic arithmetic and divisibility in the integers.

pith-pipeline@v0.9.0 · 5321 in / 1127 out tokens · 62675 ms · 2026-05-08T01:32:28.150797+00:00 · methodology

discussion (0)

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Reference graph

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